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Results tagged with sequences-and-series
Search options questions only
not deleted
user 93724
for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.
2
votes
0
answers
117
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Solving a system of differential-like equations for reverse Euler-Maclaurin summation
Aim
A particular instance of a rational zeries that has as of yet not been evaluated is:
\begin{align}
Z:= \sum_{n=1}^{\infty} \frac{\zeta(2n)}{(2n)!}. \label{EM1} \tag{EM1}
\end{align}
This sum c …
- 4,829
2
votes
0
answers
150
views
What rational zeta series with non-integer arguments appear in mathematics?
Background
Rational zeta series are series of the form $$\sum_{n=2}^{\infty} q_{n} \zeta(n + p, m), \label{1} \tag{1} $$ where $\zeta(x,m)$ is the Hurwitz zeta function and $q_{n}, \ p \in \mathbb{Q} …
- 4,829
1
vote
1
answer
97
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Finding $\lim_{n \to \infty} \sum_{k=2}^{n-2} \zeta(k) \zeta(n-k) x^{k-1} = x^{-1} - \psi_{0...
In equation (130) of this page, the identity $$\lim_{n \to \infty} \sum_{k=2}^{n-2} \zeta(k) \zeta(n-k) x^{k-1} = x^{-1} - \psi_{0}(-x) - \gamma \label{1} \tag{1} $$ is stated. Here, $\zeta(\cdot)$ is …
- 105k
11
votes
1
answer
453
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References on infinite series involving the tetration operator, like $ \sum_{n=1}^{\infty} \...
I wonder whether there are any references on infinite series involving the tetration operator, including:
\begin{align} S_{1} &:= \sum_{n=1}^{\infty} \frac{1}{ {^{n}2} } \\
&= \frac{1}{2} + \frac{1}{2 …
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2
votes
0
answers
230
views
Is there a theory of formal product series?
A few years ago, I asked a question on MSE about the existence of an infinite product representation of a functional square root of the sine function. No answers were given, though user conditionalMet …
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6
votes
0
answers
387
views
Is there a residue sum formula in quaternionic analysis?
In complex analysis, there is a formula involving the residues of complex functions that one can employ to find the value of certain infinite series.
If the function $f: \mathbb{C} \to \mathbb{C} $ sa …
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6
votes
2
answers
905
views
Has the "partial Sophomore's Dream function" been studied before?
We can consider the generalized Harmonic numbers $$H_{n,m} := \sum_{k=1}^{n} \frac{1}{k^{m}} $$ as a partial version of the Riemann zeta function, because $$\lim_{n \to \infty} H_{n,m} = \zeta(m). $$
…
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1
vote
0
answers
202
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Questions about iterating the Euler-Maclaurin summation formula
Introduction
The Euler–Maclaurin summation formula is as follows for a positive integer $p$ and a continuous function $f(\cdot)$ that is $p$ times continuously differentiable on the interval $[m,n]$ : …
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4
votes
1
answer
242
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Are there any extensive treatments on rational zeta series?
I've been trying to find an extensive, in-depth treatment of rational zeta series. Via the Wikipedia article on the topic, I've found two articles on this subject. While they are certainly very inform …
- 4,829
1
vote
0
answers
152
views
Pairs of functions with $\sum_{n} (f \circ g)(n) = \sum_{n} (g \circ f)(n) $
I was wondering there there are any pairs of functions $(f,g)$ such that $$\sum_{n=1}^{\infty} (f \circ g)(n) = \sum_{n=1}^{\infty} (g \circ f)(n) $$ on condition that they're not commutative with re …
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2
votes
1
answer
315
views
Are there any published studies on cases of infinite sums for which the Euler–Maclaurin summ...
The Euler–Maclaurin summation formula is as follows: $$\sum_{i=m}^{n} f(i) = \int_{m}^{n} f(x) dx + \frac{f(n)+f(m)}{2} + \sum_{k=1}^{\lfloor p/2 \rfloor} \frac{B_{2k}}{(2k)!}\big{(}f^{(2k-1)}(n)-f^{( …
- 12.9k
10
votes
1
answer
726
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What is known about sums of the form $\sum_{n=2}^{\infty}[\zeta(n)-1]^{p} $?
A fair bit is known about rational zeta series. This includes identities like $$ \sum_{n=2}^{\infty} [\zeta(n) -1] = 1 . $$
Many more identities can be found in articles by e.g. Borwein and Adamchik & …
- 4,829
2
votes
0
answers
154
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How can collections of rational zeta series that are equal to $\sum_{n=2}^{\infty} (\zeta(n)...
It has been discovered long ago that
$$\sum_{n=2}^{\infty} \big(\zeta(n) - 1\big) = 1. \label{1} \tag{1} $$ More recently, a generalization of this result with binomial coefficients has been obtained: …
- 6,377
2
votes
0
answers
91
views
Rational zeta series and differential-difference equations
In an earlier question, I mentioned I was looking for generalizations of $$\sum_{n=k+2}^{\infty} \binom{n-1}{k} (\zeta(n) -1) =1. \qquad \qquad (1) $$
A variation of the above identity arises by cons …
- 4,829
3
votes
1
answer
329
views
A question on an identity relating certain sums of Harmonic numbers
In the description of this question, it was established that \begin{align} \sum_{n=2}^{\infty} (\zeta(n)^{2}-1) &=
\frac{7}{4} - \zeta(2) + 2 \sum_{m=1}^{\infty} \frac{H_{m-1- \frac{1}{m}} - H_{- \f …
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